A non-standard finite difference method for a hepatitis B virus infection model with spatial diffusion

Abstract

A non-standard finite difference method is proposed for an epidemic model which describes the hepatitis B virus infection with spatial dependence. Using the theory of M-matrix, it is shown that the proposed method is unconditionally positive. Moreover, this method preserves all constant steady-state solutions of the corresponding continuous system. Through constructing discrete Lyapunov functions, the globally asymptotical stabilities of the steady-state solutions are fully determined by the basic reproduction number R 0 independent of the time and space step sizes, which coincides with the continuous system. Finally, numerical experiments are provided to illustrate the theoretical results.

Dynamic consistent NSFD scheme for a viral infection model with cellular infection and general nonlinear incidence

“…More recently, Qin et al [38] applied the NSFD scheme to discretized system (2) and showed that the discrete model has the same dynamics as the original system. Hence, following the idea of [29,38], by applying the NSFD scheme to system (3) we obtain the following discrete system:…”

“…One important advantage of Mickens' scheme is that it performs well in preserving the major dynamical properties of the approximated original continuous models (see [20,[35][36][37][38]). More recently, Qin et al [38] applied the NSFD scheme to discretized system (2) and showed that the discrete model has the same dynamics as the original system. Hence, following the idea of [29,38], by applying the NSFD scheme to system (3) we obtain the following discrete system:…”

“…However, since a discrete model generally can exhibit more complicated dynamical behavior than continuous models such as bifurcations and chaos (see [25,28] and references therein). Fortunately, a nonstandard finite difference scheme (NSFD) has been proposed by Mickens [29] and received much attention (see [20,[30][31][32][33][34][35][36][37][38] and references therein). One important advantage of Mickens' scheme is that it performs well in preserving the major dynamical properties of the approximated original continuous models (see [20,[35][36][37][38]).…”

Numerical Analysis of the Susceptible Exposed Infected Quarantined and Vaccinated (SEIQV) Reaction-Diffusion Epidemic Model

“…In epidemic models, population dynamics and population size cannot be negative, so the numerical technique must be a positivity-preserving technique. Various authors have used different positivitypreserving numerical techniques for the approximate solution of epidemic models: see, for example [14][15][16][17][18][19][20][21][22].…”

Dynamic Consistent NSFD Scheme for a Delayed Viral Infection Model with Immune Response and Nonlinear Incidence

“…However, how to select a proper discrete method so that the global properties of solutions of the corresponding continuous models can be efficiently preserved is still an open problem [14]. Recently, Mickens has made an attempt in this regard, by proposing a robust nonstandard finite difference (NSFD) scheme [15,16], which has been widely employed in the study of different kinds of epidemic models and one important advantage of Mickens' method is that it can be more efficient in preserving the global dynamics to the corresponding continuous epidemic models [10,[17][18][19][20][21][22][23]. However, there is no result about discrete viral infection model with time delays and immune response.…”